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\title{\huge Sine \footnote{This file is from the 3D-XploreMath project. \hfil\break Please see http://rsp.math.brandeis.edu/3D-XplorMath/index.html}}
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\LARGE


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\centerline{\includegraphics{sine.png}}

All trignometric functions sine, cosine, tangent, secant, cosecant, cotangent can all be simply defined in terms of a single function sine. Sine, as associated with trignometry, began in early civilization as a very important measuring science. When the function concept and calculus and analytic geometry were introduced in about 1700, sine became a function and has little to do with triangles. The sine function appears unexpectedly throughout analysis, because in essence it captures the idea of a wave, a fundamental concept in physics.

Excerpt from Robert C. Yates (1974):

\begin{quote}
Trigonometry seems to have been developed, with certain traces of Indian influence, first by the Arabs about 800 as an aid to the solution of astronomical problems. From them the knowledge probably passed to the Greeks. Johann M\"{u}ller (c.1464) wrote the first treatise: ``De triangulis omnimodis''; this was followed closely by others.
\end{quote}


% dick, how to do italics? the book title above needs italics. I'm still learning TeX, quite tedious in constant looking up for things... Help me here and i'll remember. thanks.


Sinusoid is the curve of the sine function. Sine is sometimes called circular function because the essential feature of the sine function can be thought of as a point moving around a circle in a uniform way, and the value of sine being the height of the point.

Step by step description:

1.	Draw a circle of radius one centered on the origin. 

2.	Let P be any point on the circle. Let the coordinate of P by $(p_1,p_2)$. 

3.	Let $\theta$ be the angle (in radians) formed by the three points (1,0), (0,0), P. 

4.	Sinusoid is the locus of points with coordinates ($\theta$, $p_2$).

\centerline{\includegraphics{sinusoidGen.png}}

If we limit $\theta$ to $0 < t < \pi/2$, the above becomes the classical definition of sine as the ratio of a right triangle (1,0),(0,0),P, and $sin(\theta)$ is the the heght $p_2$ divided by the hypotenuse length[(0,0),P].

The other trig functions can be defined in terms of sine, as follows.

\begin{eqnarray*}
&&\csc(\theta)=1/\sin(\theta)\\
&&\cos(\theta)=\sin(\theta+ \pi/2)\\
&&\sec(\theta)=1/\cos(\theta)\\
&&\tan(\theta)=\sin(\theta)/\cos(\theta)\\
&&\cot(\theta)=1/\tan(\theta)\\
\end{eqnarray*}

If a right triangle is place in the Cartesian coordinate system such that it lies in the first quadrant, the right angle vertex lies on the x-axes and the hypotenuse touches the origin (0,0), and if r denote (the length of) the hypotenuse, x the bottom side, y the verticle side, $\theta$ the angle of x and r, then we have the following formulas:

\begin{eqnarray*}
&&\sin(\theta) = y/r\\
&&\cos(\theta) = x/r\\
&&\tan(\theta) = y/x\\
\end{eqnarray*}


Here are some other common identities that are less obvious:

Pythagorean

\begin{eqnarray*}
\sin(x)^2 + \cos(x)^2 = 1
\end{eqnarray*}


Sum

\begin{eqnarray*}
&&\sin(a) + \sin(b) = 2 \sin((a+b)/2) \cos((a-b)/2)\\
&&\sin(a) - \sin(b) = 2 \cos((a+b)/2) \sin((a-b)/2)\\
&&\cos(a) + \cos(b) = 2 \cos((a+b)/2) \cos((a-b)/2)\\
&&\cos(a) - \cos(b) = -2 \sin((a+b)/2) \sin((a-b)/2)\\
\end{eqnarray*}


Addition, subtraction, Double-angle
\begin{eqnarray*}
&&\sin(a+b) = \cos(a) \sin(b) + \cos(b) \sin(a) \\
&&\cos(a+b) = \cos(a) \cos(b) - \sin(a) \sin(b)\\
&&\tan(a+b) =(\tan(a)+\tan(b))/(1-\tan(a) \tan(b))\\
\end{eqnarray*}


Product

\begin{eqnarray*}
&&\sin(a) \cos(b) = (\sin(a+b) + \sin(a-b))/2\\
&&\cos(a) \cos(b) = (\cos(a+b) + \cos(a-b))/2\\
&&\sin(a) \sin(b) = (\cos(a-b) - \cos(a+b))/2\\
\end{eqnarray*}


Half-angle
(Sign must be chosen)

\begin{eqnarray*}
\sin(x/2) &&= +- \sqrt{((1-\cos(x))/2}\\
\cos(x/2) &&= +-\sqrt{(1+\cos(x))/2}\\
\tan(x/2) &&= +-\sqrt{(1-\cos(x))/(1+\cos(x))} \\
&&= \sin(x)/(1+\cos(x)) \\
&&= (1-\cos(x))/\sin(x)
\end{eqnarray*}


Inverse Identities

\begin{eqnarray*}
\mathrm{arcsin}(-x) &=& -\mathrm{arcsin}(x) \\
\mathrm{arccos}(x) + \mathrm{arccos}(-x) &=& \pi \\
\mathrm{arcsin}(x) + \mathrm{arccos}(x) &=& \pi/2 \\
\mathrm{arcsin}(x) &=& \mathrm{arccsc}(1/x) \\
\mathrm{arccos}(x) &=& \mathrm{arcsec}(1/x) \\
\mathrm{arctan}(x) &=& \mathrm{arccot}(1/x) \\
\end{eqnarray*}


Laws of sine and Laws of cosine

For any triangle with side lengths a, b, and c, whose opposite angles are $\alpha, \beta, and \gamma $ respectively, we have:

\begin{eqnarray*}
&& \sin(\alpha)/a = \sin(\beta)/b = \sin(\gamma)/c  \\
&& a^2 = b^2 + c^2 - 2 b c \cos(\alpha) \\
\end{eqnarray*}



\section{trivia}

\subsection{Orthogonal Projection of Helix}
\centerline{\includegraphics{helicoid_projection.png}}

Sinusoid is the orthogonal projection of the space curve helix. In 3DXM, helix can be seen under the Space Curves category.

\subsection{development of cut cylinder}
\centerline{\includegraphics{sinusoidCylinderCut.png}}

Sinusoid is the development of an obliquely cut right circular cylinder. (the edge of the cylinder rolled out is a sinusoid).



XL.


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